Topological degree of solution mappings in functional and ordinary differential equations
Initial value problems for functional- differential equations \[ x'(t)=f(t,x_t), \quad x_0=\phi , \tag{1} \] with \(x_t(s)=x(t+s)\) for \(s\in [-r,0]\) and \(\phi \in C[-r,0]\), are considered. Function \(f\) is assumed to satisfy the following condition: there exist two positive constants \(a\) and \(b\) such that \(| f(t,\psi)| \leq a+b| \psi | \) for \((t,\psi)\in [0,T]\times C[-r,0]\). This assumption ensures that every solution to (1) is extendable to be continuous up to \(t=T\). The mapping \(\Gamma =\{x_T: x\) is a solution to (1) defined on \([-r,T]\}\) is multivalued and acts from \(C[-r,0]\) to \(C[-r,0]\). A topological degree for the mapping \(\Gamma\) is defined. It is shown that in case \(r=0\) this definition is equivalent to the definition of the topological degree for ordinary differential equations, which was introduced formerly by the author.
scientific article; zbMATH DE number 1140486